Understanding filtration › 1. Separation basics

CHAPTER 1.3

Reading a filtration curve

A funnel, a graduated cylinder and a stopwatch are enough to tell whether a sludge filters well, whether the filter cake or the cloth is holding back the flow, and whether raising the pressure will help. You still need to know how to read the straight line you get.

Nicolas Ract · 8 min read · October 2026

The starting point: Darcy's law

When a liquid flows slowly through a porous medium, the pressure loss is proportional to the flow velocity and to the viscosity of the liquid, and inversely proportional to the permeability of the medium: this is Darcy's law [3]. In filtration on a medium, the liquid passes through two resistances in series: that of the cake and that of the cloth [3]. The filtrate flow rate is then written:

• V: volume of filtrate collected since the start (m³); t: time (s);
• A: filtration area (m²); ΔP: pressure difference across cake and cloth (Pa);
• μ: filtrate viscosity (Pa·s);
• w: mass of dry solids deposited per m³ of filtrate (kg/m³), taken in practice as the sludge concentration [1];
• α: specific cake resistance (m/kg); R: medium resistance (m⁻¹) [4] [5].

The term α × w × V ÷ A is the cake resistance: it increases as the cake grows thicker. The term R is the cloth, together with the particles lodged in it. At the start of the cycle the cloth dominates; after that, the cake does.

At constant pressure: a straight line

If the pressure stays constant during the test, and if the specific cake resistance does not change, the equation can be integrated to give [4] [5]:

Plotting t/V on the y-axis and V on the x-axis therefore gives a straight line, often called the Ruth line [4]:

• the slope gives the specific cake resistance: α = 2 × slope × A² × ΔP ÷ (μ × w);
• the intercept gives the medium resistance: R = intercept × A × ΔP ÷ μ; if it is close to zero, the cloth hardly restricts the flow at all [4].

Not all of the plot is a straight line. At the very beginning the points often deviate: the cake has not yet formed and, at low pressure or high concentration, settling of the sludge distorts the measurements. At the end of the test, once the cake has formed and starts to drain, the curve turns upwards. Only the straight portion is used [1] [5].

The laboratory test

Under vacuum, on a Büchner funnel

Degrémont's Water Handbook (Mémento technique de l'eau) describes the test as follows [1]:

• a Büchner funnel fitted with fast filter paper, placed on a graduated cylinder connected to the vacuum;
• 100 to 150 mL of prepared sludge (flocculated or conditioned as on the plant), enough to form a cake of 8 to 10 mm;
• vacuum applied, of the order of 0.5 bar, then the stopwatch started;
• the filtrate volume read every 10 to 60 seconds depending on the rate, until the cake cracks and the vacuum drops.

Under pressure, in a filtration cell

To get closer to filter press conditions, the same source describes a closed cell into which 100 to 150 mL of sludge is poured and left to stand for 15 seconds to form a precoat, before a pressure of between 0.1 and 15 bar is gradually applied. The filtrate volume is then recorded against time and the resistance calculated in the same way [1].

Measuring compressibility

By repeating the test at several pressures, for example 49, 147, 441 and 1,323 kPa (about 0.5, 1.5, 4.5 and 13 bar), the logarithm of the specific resistance is plotted against the logarithm of the pressure. The slope of this line is the compressibility coefficient [1], whose interpretation is given in chapter 1.2. A zero slope indicates an incompressible cake; the steeper it is, the less useful additional pressure becomes.

Some orders of magnitude

According to the Degrémont Handbook, the specific resistance measured at 0.5 bar for an unconditioned mixed municipal sludge is of the order of 10¹⁴ to 10¹⁵ m/kg; it must be brought below 10¹² m/kg by conditioning for the sludge to dewater properly on a filter press. Decarbonation sludges, of the order of 10¹¹ to 5 × 10¹¹ m/kg, can be sent to it without conditioning [2]. Above all, these values illustrate the breadth of the scale: a factor of a thousand or more separates a sludge that filters well from one that does not.

What it tells you in practice

• Setting the conditioning. The Büchner test is used to find the optimum reagent dose [1]: you look for the dose that lowers the slope the most, and beyond which it no longer falls.
• Choosing the operating pressure. The compressibility coefficient tells you whether a higher pressure will shorten the cycle or not [2].
• Knowing what is holding things back. A steep slope points to the cake: act on the sludge (conditioning, thickening). A high intercept points to the medium: cloth too tight or already blinded (see chapter 4.4).
• Comparing cloths. Field experience: replacing the filter paper with a disc cut from the cloth, on the same sludge and at the same pressure, the slope should stay roughly the same; it is the intercept, the clarity of the first filtrate and the cake discharge that separate the cloths. A disc of used cloth, compared with a new disc, measures its blinding. These tests give direction, but do not replace a comparative trial on the machine (see chapter 6.2).

The limits of the exercise

The model assumes constant pressure, a homogeneous cake and a fixed cloth resistance during the test. On a real filter press, the pressure rises gradually during filling, the cake forms on two faces and may compact; on a vacuum filter, it drains and cracks. A small laboratory test therefore gives reliable trends and useful comparisons, but extrapolating it directly to an industrial cycle time calls for caution and, ideally, a pilot trial.